spectral and algebraic instabilities in thin keplerian disks: i – linear theory
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Spectral and Algebraic Instabilities in Thin Keplerian Disks: I – Linear Theory. Edward Liverts Michael Mond Yuri Shtemler. Accretion disks – A classical astrophysical problem. - PowerPoint PPT PresentationTRANSCRIPT
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Spectral and Algebraic Instabilities in Thin Keplerian Disks: I – Linear
Theory
Edward Liverts
Michael Mond
Yuri Shtemler
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Accretion disks – A classical astrophysical problem
Original theory is for infinite axially uniform cylinders. Recent studies
indicate stabilizing effects of finite small thickness. Coppi and Keyes,
2003 ApJ., Liverts and Mond, 2008 MNRAS have considered modes
contained within the thin disk.
Weakly nonlinear studies indicate strong saturation of the instability.
Umurhan et al., 2007, PRL.
Criticism of the shearing box model for the numerical simulations and its
adequacy to full disk dynamics description. King et al., 2007 MNRAS,
Regev & Umurhan, 2008 A&A.
Magneto rotational instability (MRI) Velikhov, 1959, Chandrasekhar,
1960 – a major player in the transition to turbulence scenario, Balbus
and Hawley, 1992.
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Algebraic Vs. Spectral Growth
spectral analysis
ni tnt
f t f e ( , ) ( ) r r
Spectral stability if nIm 0
- Eigenvalues of the linear operatorn
for all s n
Short time behavior
1000eR
f
f tG t Max
f(0)
( )( )
(0)
Transient growth for Poiseuille flow.
Schmid and Henningson, “Stability and Transition in Shear Flows”
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To investigate the linear as well as nonlinear
development of small perturbations in realistic thin disk
geometry.
Goals of the current research
Derive a set of reduced model nonlinear equations
Follow the nonlinear evolution.
Have pure hydrodynamic activities been thrown out too hastily ?
Structure of the spectrum in stratified (finite) disk. Generalization to
include short time dynamics. Serve as building blocks to nonlinear
analysis.
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Magneto Rotational Instability (MRI)
Governing equations
.00,v
,4
,v-v
,1v 2
2
Bt
Bc
jBBdt
Bd
Bjc
rc
dt
d s
tzbbzB ,,v,v,
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z
R0
H0
Thin Disk Geometry
Vertical stretching
z
0
0
H
R
1
z
Supersonic rotation:
1sM
r
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2 2
2
dP
dt n
Vj B
Vertically Isothermal Disks
( , ) ( , ) ( ) P r n r T r
*2*
4 P
B
Vertical force balance:2 2/2 ( )( , ) ( ) H rn r N r e
Radial force balance:3/2( ) ( ) , (r)=V r r r r
Free functions: The thickness is determined by
.
( )T r( )H r ( ) ( ) / ( )H r T r r
( )N r
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• I. Dominant axial component • II. Dominant toroidal component
Two magnetic configurations
0 ( , )
( , )
z zB B r
B B r
0 0
( , )
( , )
z zB B r
B B r
( )B r
Magnetic field does not influence lowest order equilibrium
but makes a significant difference in perturbations’
dynamics.
Free function: ( )zB r Free function: ( )B r
8
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Linear Equations: Case Iz
zBzB2 2/2 ( )( , ) ( ) H rn r N r e
Self-similar
variable
( )
H r
0B Consider
2 2
2
( ) ( ) ( )( )
( )
z
N r H r rr
B r
2 2 2( ) ( ) ( )sH r r C r
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2
( ) 12 ( ) 0
( ) ( )
1 ( ) 10
2 ( ) ( )
( ) 0
ln( ) 0
ln
r r
r
r r
r
V Brr V
t r n
V BrV
t r n
B Vr
t
B V dr B
t d r
The In-Plane Coriolis-Alfvén System
2 /2( ) n e
2 2
2
( ) ( ) ( )( )
( )
z
N r H r rr
B r
No radial derivatives Only spectral instability possible
Boundary conditions
, 0 rB
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The Coriolis-Alfvén System
n2 /2( ) n e
2( ) sec ( ) n h b
22 /2sec ( ) h b d e d
2
b
Approximate profile
Change of independent
variabletanh( ) b
1
2
Spitzer, (1942)
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The Coriolis-Alfvén Spectrum
,ˆ( )( ) 0 rK K V L L
2[(1 ) ]
d d
d d L
Legendre operator
2 22
( )3 2 9 16
2
rK
b
ˆ( , , ) ( , ) , ( ) ( )f r t f r e r r i t
,ˆ ( ) r kV P ( 1) , 1,2,K k k k
3
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The Coriolis-Alfvén Spectrum
13
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The Coriolis-Alfvén Dispersion Relation
2 22
( )3 2 9 16
2
rK
b ( 1)k k
2 4 2 ( )ˆ ˆ ˆ ˆ ˆ( 6) 9(1 ) 0 , cr
r
2
( 1)3
crb
k k
Universal condition for
instability:
ˆ 1 cr
Number of unstable modes increases with .
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The Coriolis-Alfvén Dispersion Relation-The MRI
Im , Re Im
( 1) 0.42cr k max ˆIm for 3
Im
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( ) 0( )
( ) ( ) 0
z
z
V nr
t n
nr n V
t
ˆ ˆ( ) ( 1) 0n n
The Vertical Magnetosonic System
2 22
2 2
ˆˆ(1 ) 2 0
1
d nn
d
ˆ( , , ) ( , )
( ) ( )
f r t f r e
r r
i t
2( ) sec ( ) n h b
tanh( ) b
16
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Continuous acoustic spectrum
21 1 2 2( ) 1 ( ) ( )n c f c f
2
11
( ) ( )1
f
2
21
( ) ( )1
f
2 21
Define: 2( ) 1 ( )n g
2 2 22
2 2 2
1(1 ) 2 2 0
1
d g d gg
d d
Associated Legendre
equation
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Continuous eigenfunctions
1( )f
2 ( )f
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Linear Equations: Case II z
B
( , )
( , )
z zB B r
B B r
2 2/2 ( )( , ) ( ) H rn r N r e
Axisymmetric perturbations are spectrally stable.
Examine non-axisymmetric perturbations.
( ) , - , , ( ) ( )
r drr t
H r H r
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Two main regimes of perturbations dynamics
I. Decoupled inertia-coriolis
and MS modes
II. Coupled inertia-coriolis
and MS modes
A. Inertia-coriolis waves driven
algebraically by acoustic
modes
B. Acoustic waves driven
algebraically by inertia-coriolis
modes
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I.A – Inertial-coriolis waves driven by MS modes
1 10
( ) ( )
[ ( ) ]10
( )
z
z
z
bV n
n
n Vn
n
b V
2
2
( ) ( )( )
( )sN c
B
The magnetosonic
eigenmodes
ˆ( , , , ) ( , ) i ikf f e
22
2
ˆ ˆ( ) 1 ( ) ˆ( 2) 0( ) 1 ( ) 1
d b n db nb
n d nd
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WKB solution for the eigenfunctions
Turning points at *
The Bohr-Sommerfeld
condition*
2 2
*
1( ) ( 1) , ( ) ( * )
2 4d m
1MS m ˆ( , , , ) ( , ) i ikf f e
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The driven inertia-coriolis (IC) waves
1 0
0
2 { , , } { , , }
1{ , , }
2
rz z
r z
VV L n V b L n V b
VV K n V b
IC operator Driving MS
modes
2
ˆ[ ( ) ]1 1 1ˆˆ( ) ( ) exp[ ]( ) ( )1
MS zr MS
MS
i d vv D ik b i ik
d
1
( )r
rdv
bdS
ln( )
dD
d
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II – Coupled regime
Need to go to high axial and azimuthal wave numbers
1, zk k
22
2
12 2 ( ) ( ) {[ ( ) 1] ( ) }r z r
r r r r r zd b db db
k k k k b k k bd dd
22
2
1[(1 ) ( ) ]z
z z r rd b
k k b k k bd
3( , ) ( )
2r r rk k D k
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Instability
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Hydrodynamic limit
Both regimes have clear hydrodynamic limits.
Similar algebraic growth for decoupled regime.
Much reduced growth is obtained for the coupled regime.
Consistent with the
following hydrodynamic
calculations:
Umurhan et al. 2008
Rebusco et al. 2010
Shtemler et al. 2011
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II – Nonlinear theory
),()1((4
1)1(
22
1
Ov
vv
b
bb
v
vL
v
v
tr
zr
zr
Ar
),()1( 22
Ob
bv
v
vb
b
bL
b
b
tr
zr
zr
Ar
),(1
)1(8
11
22
22
O
v
bbvL
v
t z
rzs
z
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Weakly nonlinear asymptotic model
.)(,,3
21427
cc i
~A
Ginzburg-Landau equation
3AAt
A
Landau, (1948)
Phase plane
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Numerical Solution
...)()()()()()(),( 3)3(
,2)2(
,1)1(,, yPtvyPtvyPtvytv zzzz
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Weakly nonlinear theory, range I
.)(,,3
2,04])3(2][2[ 14
27222cc i
)].()([)(
)()(),(
1
)()( 20
218
74
32
176
yPyPtAb
byPtA
v
v
c
crcr
)561)((
)1()(42
2
tC
tBvz
).(10
),(9
4
),(35
8
3
32
4
AOB
dt
dC
AOACdt
dB
AOABAdt
dA
nmmn ndPP
1
1 12
2)()(
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Nonlinear theory, range II
)()()()(
),()()()(
4422
3311
PtCPtC
PtBPtBvz
).(7
20
),(5
6
5
1
),(13
35
52
245
104
315
),(17
49
52
105
104
385
),()175
8
75
8(
33
4
331
2
3242
3
3242
1
321
AOBdt
dC
AOBBdt
dC
AOACCdt
dB
AOACCdt
dB
AOBBAAdt
dA
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Generation of toroidal magnetic field
)]()()[( 20 PPtAb
...)()()()()()(),( 3)3(
2)2(
1)1( yPtbyPtbyPtbytb
rtdr
dbb r
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Conclusions
It is shown that the spectrum in a stratified disk (the finite thickness) contains a discrete part and a continuous one for AC and MS waves respectively. In the linear approximation, the AC and MS waves are decoupled. There is a special critical value of parameter plasma beta for which the system becomes unstable if
Both main regimes of perturbations dynamics, namely decoupled (coupled) AC and MS modes, are discussed in detail and compared with hydrodynamic limits.
It was found from numerical computations that the generalization to include nonlinear dynamics leads to saturation of the instability.
The nonlinear behavior is modeled by Ginzburg-Landau equation in case of weakly nonlinear regime (small supercriticality) and by truncated system of equations in the case of fully nonlinear regime. The latter is obtained by appropriate representation of the solutions of the nonlinear problem using a set of eigenfunctions for linear operator corresponding to AC waves
c0c